College Physics

Raymond A. Serway, Jerry S. Faughn, Chris Vuille

Chapter 3

Vectors and Two-Dimensional Motion - all with Video Answers

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Chapter Questions

Problem 1

Vector $\vec{A}$ has a magnitude of 29 units and points in the positive $y$ -direction. When vector $\overrightarrow{\mathbf{B}}$ is added to $\overrightarrow{\mathrm{A}}$, the resultant vector $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathbf{B}}$ points in the negative $y$ -direction with a magnitude of 14 units. Find the magnitude and direction of $\overrightarrow{\mathbf{B}}$.

Darren Wilson

Darren Wilson

Numerade Educator

Problem 2

Vector $\vec{A}$ has a magnitude of $8.00$ units and makes an angle of $45,0^{\circ}$ with the positive $x$ -axis. Vector $\overrightarrow{\mathbf{B}}$ also has a magnitude of $8.00$ units and is directed along the negative $x$ -axis. Using graphical methods, find (a) the vector $\operatorname{sum} \mathrm{A}+\overrightarrow{\mathrm{B}}$ and (b) the vector difference $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$.

Ryan Williams

Numerade Educator

Problem 3

Vector $\overrightarrow{\mathrm{A}}$ is $3.00$ units in length and points along the positive $x$ -axis. Vector $\overrightarrow{\mathbf{B}}$ is $4.00$ units in length and points along the negative $y$ -axis. Use graphical methods to find the magnitude and direction of the vectors (a) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathbf{B}}$ and
(b) $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathbf{B}}$

Averell Hause

Carnegie Mellon University

Problem 4

Each of the displacement vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in Figure P3.4 has a magnitude of $3.00 \mathrm{~m}$. Graphically find
(a) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathbf{B}}$
(b) $\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$
(c) $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}$, and
(d) $\overrightarrow{\mathbf{A}}-2 \overrightarrow{\mathbf{B}}$.

Ryan Williams

Numerade Educator

Problem 5

A roller coaster moves $200 \mathrm{ft}$ horizontally and then rises $135 \mathrm{ft}$ at an angle of $30.0^{\circ}$ above the horizontal. Next, it travels $135 \mathrm{ft}$ at an angle of $40.0^{\circ}$ below the horizontal. Use graphical techniques to find the roller coaster's displacement from its starting point to the end of this movement.

Ryan Williams

Numerade Educator

Problem 6

An airplane flies $200 \mathrm{~km}$ due west from city $\mathrm{A}$ to city $B$ and then $300 \mathrm{~km}$ in the direction of $30.0^{\circ}$ north of west from city $\mathrm{B}$ to city $\mathrm{C}$. (a) In straight-line distance, how far is city $C$ from city $A ?$ (b) Relative to city $A$, in what direction is city C? (c) Why is the answer only approximately correct?

Supratim Pal

Numerade Educator

Problem 7

A plane flies from base camp to lake $A$, a distance of $280 \mathrm{~km}$ at a direction of $20.0^{\circ}$ north of east. After dropping off supplies, the plane flies to lake $\mathrm{B}$, which is $190 \mathrm{~km}$ and $30.0^{\circ}$ west of north from lake A. Graphically determine the distance and direction from lake $\mathrm{B}$ to the base camp.

Matthew Baker

Numerade Educator

Problem 8

A jogger runs $100 \mathrm{~m}$ due west, then changes direction for the second leg of the run. At the end of the run, she is 175 In away from the starting point at an angle of $15.0^{-}$ north of west. What were the direction and length of her second displacement? Use graphical techniques.

Ryan Williams

Numerade Educator

Problem 9

A man lost in a maze makes three consecutive displacements so that at the end of his travel he is right back where he started. The first displacement is $8.00 \mathrm{~m}$ westward. and the second is $13.0 \mathrm{~m}$ northward. Use the graphical method to find the magnitude and direction of the third displacement.

Ryan Williams

Numerade Educator

Problem 10

The magnitude of vector $\vec{A}$ is $35.0$ units and points in the direction $325^{\circ}$ counterclockwise from the positive $x$ -axis. Calculate the $x$ -and $y$ -components of this vector.

Averell Hause

Carnegie Mellon University

Problem 11

A golfer takes two putts to get his ball into the hole once he is on the green. The first putt displaces the ball $6.00 \mathrm{~m}$ east. the second $5.40 \mathrm{~m}$ south. What displacement would have been needed to get the ball into the hole on the first putt?

Ryan Williams

Numerade Educator

Problem 12

$\mathrm{A}$ figure skater glides along a circular path of radius $5.00 \mathrm{~m}$. If she coasts around one half of the circle, find
(a) the magnitude of the displacement vector and (b) what distance she skated. (c) What is the magnitude of the displacement if she skates all the way around the circle?

Ryan Williams

Numerade Educator

Problem 13

$\mathrm{A}$ figure skater glides along a circular path of radius $5.00 \mathrm{~m}$. If she coasts around one half of the circle, find
(a) the magnitude of the displacement vector and (b) what distance she skated. (c) What is the magnitude of the displacement if she skates all the way around the circle?

Matthew Baker

Numerade Educator

Problem 14

A hiker starts at his camp and moves the following distances while exploring his surroundings: $75.0 \mathrm{~m}$ north. $2.50 \times 10^{2} \mathrm{~m}$ east, $125 \mathrm{~m}$ at an angle $30.0^{\circ}$ north of east. and $1.50 \times 10^{2} \mathrm{~m}$ south. (a) Find his resultant displacement from camp. (Take east as the positive $x$ -direction and north as the positive $y$ -direction.) (b) Would changes in the order in which the hiker makes the given displacements alter his final position? Explain.

Nicholas Mogoi

Numerade Educator

Problem 15

A vector has an $x$ -component of $-25.0$ units and a $y$ component of $40.0$ units. Find the magnitude and direction of the vector.

Averell Hause

Carnegie Mellon University

Problem 16

A quarterback takes the ball from the line of scrimmage. runs backwards for $10.0$ yards, then runs sideways parallel to the line of scrimmage for $15.0$ yards. At this point, he throws a $50.0$ -yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football's resultant displacement?

Ryan Williams

Numerade Educator

Problem 17

The cye of a hurricane passes over Grand Bahama Island in a direction $60.0^{\circ}$ north of west with a speed of $41.0 \mathrm{~km} / \mathrm{h}$. Three hours later the course of the hurricane suddenly shifts due north, and its speed slows to $25.0 \mathrm{~km} / \mathrm{h}$. How far from Grand Bahama is the hurricane
4. $50 \mathrm{~h}$ after it passes over the island?

Ryan Williams

Numerade Educator

Problem 18

A small map shows Atlanta to be 730 miles in a direction $5^{\circ}$ north of east from Dallas. The same map shows that Chicago is 560 miles in a direction $21^{n}$ west of north from Atlanta. Assume a flat Earth and use the given information to find the displacement from Dallas to Chicago.

Ryan Williams

Numerade Educator

Problem 19

commuter airplane starts from an airport and takes the route shown in Figure $\mathrm{P} 3.19 .$ The plane first flies to city $A$, located $175 \mathrm{~km}$ away in a direction $30.0^{\circ}$ north of cast. Next, it flies for $150 \mathrm{~km} 20.0^{\circ}$ west of north, to city $B$. Finally, the plane flies $190 \mathrm{~km}$ due west, to city $C$. Find the location of city Crelative to the location of the starting point.

Ryan Williams

Numerade Educator

Problem 20

The helicopter view in Figure $\mathrm{P} 3.20$ shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force a third person would have to exert on the mule to make the net force equal to zero. The forces are measured in units of newions (N).

Ryan Williams

Numerade Educator

Problem 21

A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are $4.00 \mathrm{~m}$ to the north, $2.00 \mathrm{~m} 45.0^{\circ}$ north of east, and $1.00 \mathrm{~m}$ at $30.0^{\circ}$ west of south. Starting at the same initial point, an expert golfer could make the hole in what single displacement?

Ryan Williams

Numerade Educator

Problem 22

One of the fastest recorded pitches in major-league baseball, thrown by Joel Zumaya in 2006, was clocked at $101.0 \mathrm{mi} / \mathrm{h}$ (Fig. P3.22). If a pitch were thrown horizonLally with this velocity, how far would the ball fall vertically by the time it reached home plate, $60.5 \mathrm{ft}$ away?

Ryan Williams

Numerade Educator

Problem 23

A stuclent stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of $18.0 \mathrm{~m} / \mathrm{s}$. The cliff is $50.0 \mathrm{~m}$ above a flat, horizontal beach as shown in Figure P3.23 (page 78 ). (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity? (c) Write the equations for the $x$ -and $y$ -components of the velocity of the stone with time. (d) Write the equations for the position of the stone with time, using the coordinates in Figure P3.23, (c) How long after being released does the stone strike the beach

Donald Albin

Numerade Educator

Problem 24

A peregrine falcon (Fig. P3.24) is the fastest bird, flying at a speed of $200 \mathrm{mi} / \mathrm{h}$. Nature has adapted the bird to reach such a speed by placing baffles in its nose to prevent air from rushing in and slowing it down. Also, the bird's eyes adjust their focus faster than the eyes of any other creature, so the falcon can focus quickly on its prey. Assume a peregrine falcon is moving horizontally at its top speed at a height of $100 \mathrm{~m}$ above the ground when it brings its wings into its sides and begins to drop in free fall. How far will the bird fall vertically while traveling horizontally a distance of $100 \mathrm{~m}$ ?

Ryan Williams

Numerade Educator

Problem 25

The best leaper in the animal kingdom is the puma, which can jump to a height of $12 \mathrm{ft}$ when leaving the ground at an angle of $45^{\circ}$. With what speed, in SI units, must the animal leave the ground to reach that height?

Ryan Williams

Numerade Educator

Problem 26

The record distance in the sport of throwing cowpats is $81.1 \mathrm{~m}$. This record toss was set by Steye Urner of the United States in 1981 . Assuming the initial launch angle was $45^{\circ}$ and neglecting air resistance, determine
(a) the initial speed of the projectile and (b) the total time the projectile was in flight. (c) Qualitatively, how would the answers change if the launch angle were greater than $45^{\circ}$ ? Explain.

Ryan Williams

Numerade Educator

Problem 27

Atennis player standing $12.6 \mathrm{~m}$ from the net hits the ball at $3.00^{\circ}$ abowe the horizontal. To clear the net, the ball must rise at least $0.330 \mathrm{~m}$. If the ball just clears the net at the apex of its trajectory, how fast was the ball moving when it left the racket?

Ryan Williams

Numerade Educator

Problem 28

From the window of a building, a ball is tossed from a height $y_{0}$ above the ground with an initial velocity of $8,00 \mathrm{~m} / \mathrm{s}$ and angle of $20.0^{\circ}$ below the horizontal. It strikes the ground $3.00 \mathrm{~s}$ later. (a) If the base of the building is taken to be the origin of the coordinates, with upward the positive y-direction, what are the initial coordinates of the ball? (b) With the positive $x$ -direction chosen to be out the window, find the $x$ -and $y$ -components of the initial velocity. (c) Find the equations for the $x$ -and $y$ components of the position as functions of time. (d) How far horizontally from the base of the building does the ball strike the ground? (e) Find the height from which the ball was thrown. (f) How long does it take the ball to reach a point $10.0 \mathrm{~m}$ below the level of launching?

Donald Albin

Numerade Educator

Problem 29

$A$ brick is thrown upward from the top of a building at an angle of $25^{\circ}$ to the horizontal and with an initial speed of $15 \mathrm{~m} / \mathrm{s}$. If the brick is in flight for $3.0 \mathrm{~s}$, how tall is the building?

Averell Hause

Carnegie Mellon University

Problem 30

An artillery shell is fired with an initial velocity of $300 \mathrm{~m} / \mathrm{s}$ at $55.0^{\circ}$ above the horizontal. To clear an avalanche, it explodes on a mountainside $42.0 \mathrm{~s}$ after firing. What are the $x$ -and $y$ -coordinates of the shell where if explodes, relative to its firing point?

Averell Hause

Carnegie Mellon University

Problem 31

A car is parked on a cliff overlooking the ocean on an incline that makes an angle of $24.0^{\circ}$ below the horizontal. The negligent driver leaves the car in neutral, and the emcrgency brakes are defective. The car rolls from rest down the incline with a constant acceleration of $4.00 \mathrm{~m} / \mathrm{s}^{2}$ for a distance of $50.0 \mathrm{~m}$ to the edge of the cliff, which is $30.0 \mathrm{~m}$ above the ocean. Find (a) the car's position relative to the base of the cliff when the car lands in the
ocean and (b) the length of time the car is in the air.

Donald Albin

Numerade Educator

Problem 32

A fireman $50.0 \mathrm{~m}$ away from a burning building directs a stream of water from a ground-level fire hose at an angle of $30.0^{\circ}$ above the horizontal. If the speed of the stream as it leaves the hose is $40.0 \mathrm{~m} / \mathrm{s}$, at what height will the stream of water strike the building?

Narayan Hari

Numerade Educator

Problem 33

IA projectile is launched with an initial speed of $60.0 \mathrm{~m} / \mathrm{s}$ at an angle of $30.0^{\circ}$ above the horizontal. The projectile lands on a hillside $4.00 \mathrm{~s}$ later. Neglect air friction. (a) What is the projectile's velocity at the highest point of its trajectory? (b) What is the straight-line distance from where the projectile was launched to where it hits its target?

Ryan Williams

Numerade Educator

Problem 34

A soccer player kicks a rock horizontally off a $40.0$ -m-high cliff into a pool of water. If the player hears the sound of the splash $3.00 \mathrm{~s}$ later, what was the initial speed given to the rock? Assume the speed of sound in air to be $343 \mathrm{~m} / \mathrm{s}$.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 35

A jet airliner moving initially at $3.00 \times 10^{2} \mathrm{mi} / \mathrm{h}$ due east enters a region where the wind is blowing $1.00$ $\times 10^{2} \mathrm{mi} / \mathrm{h}$ in a direction $30.0^{\circ}$ north of east. (a) Find the components of the velocity of the jet airliner relative to the air, $\vec{v}_{\mu}$ (b) Find the components of the velocity of the air relative to Earth, $\vec{v}_{A E}$. (c) Write an equation analogous to Equation $3.16$ for the velocities $\vec{v}_{j}, \vec{v}_{A E}$, and $\vec{v}_{j k}$ -
(d) What is the speed and direction of the aircraft relative to the ground?

Ryan Williams

Numerade Educator

Problem 36

A boat moves through the water of a river at. $10 \mathrm{~m} / \mathrm{s}$ relative to the water, regardless of the boat's direction. If the water in the river is flowing at $1.5 \mathrm{~m} / \mathrm{s}$, how long daes it Lake the boat to make a round trip consisting of a $300-\mathrm{m}$ displacement downstream followed by a 300-m displacement upstream?

Ryan Williams

Numerade Educator

Problem 37

18 A chinook (king) salmon (genus Oncorhynchus) can jump out of water with a speed of $6.26 \mathrm{~m} / \mathrm{s} .$ (Sec Problem $4.9$, page 111 for an investigation of how the fish can leave the water at a higher speed than it can swim underwater.) If the salmon is in a stream with water speed equal to $1.50 \mathrm{~m} / \mathrm{s}$, how high in the air can the fish jump if it leaves the water traveling vertically upwards relative to the Earth?

Ryan Williams

Numerade Educator

Problem 38

A river flows due east at $1.50 \mathrm{~m} / \mathrm{s}$. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of $10.0 \mathrm{~m} / \mathrm{s}$ due north relative to the water. (a) What is the velocity of the boat relative to the shoret (b) If the river is $300 \mathrm{~m}$ wide, how far downstream has the boat moved by the time it reaches the north shore?

Ryan Williams

Numerade Educator

Problem 39

A rowboat crosses a river with a velocity of $3.30 \mathrm{mi} / \mathrm{h}$ at an angle $62.5^{\circ}$ north of west relative to the water. The river is $0.505 \mathrm{mi}$ wide and carries an eastward current of $1.25 \mathrm{mi} / \mathrm{h}$. How far upstream is the boat when it reaches the opposite shore:

Ryan Williams

Numerade Educator

Problem 40

A river has a steady speed of $0.500 \mathrm{~m} / \mathrm{s}$. A student swims upstream a distance of $1.00 \mathrm{~km}$ and swims back to the starting point. (a) If the student can swim at a speed of $1.20 \mathrm{~m} / \mathrm{s}$ in still water, how long does the trip take?
(b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current?

Yaqub Khan

Numerade Educator

Problem 41

$A$ river has a steady speed of $0.500 \mathrm{~m} / \mathrm{s}$. A student swims upstream a distance of $1.00 \mathrm{~km}$ and swims back to the starting point. (a) If the student can swim at a speed of $1.20 \mathrm{~m} / \mathrm{s}$ in still water, how long does the trip take?
(b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take lonyer when there is a current?

Ryan Williams

Numerade Educator

Problem 42

A river has a steady speed of $v_{1}$ A student swims upstream a distance $d$ and back to the starting point.
(a) If the student can swim at a speed of $v$ in still water, how much time $t_{\text {up }}$ does it take the student to swim upstream a distance d? Express the answer in terms of $d$. $v$, and $v_{c}$ (b) Using the same variables, how much time $t_{\text {down }}$ does it take to swim back downstream to the starting point? (c) Sum the answers found in parts (a) and (b) and show that the time $t_{a}$ required for the whole trip can be written as
$$
t_{d}=\frac{2 d / v}{1-v^{4} / v^{2}}
$$
(d) How much time $t_{b}$ does the trip take in still water?
(e) Which is larger, $t_{n}$ or $t_{5}^{2}$ Is it always larger?

Ryan Williams

Numerade Educator

Problem 43

A bomber is flying horizontally over level terrain at a speed of $275 \mathrm{~m} / \mathrm{s}$ relative to the ground and at an altitude of $3,00 \mathrm{~km}$. (a) The bombardier releases one bomb. How far does the bomb travel horizontally between its release and its impact on the ground? Ignore the effects of air resistance. (b) Firing from the people on the ground suddenly incapacitates the bombardier before he can call. "Bombs away!" Consequently, the pilot maintains the plane's original course, altitude, and speed through a storm of flak. Where is the plane relative to the bomb's point of impact when the bomb hits the ground? (c) The plane has a telescopic bombsight set so that the bomb hits the target seen in the sight at the moment of release. At what angle from the vertical was the bombsight set?

Ryan Williams

Numerade Educator

Problem 44

A moving walkway at an airport has a speed $v_{1}$ and a lengih $1 .$ A woman stands on the walkway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the walkway with a speed of $\tau_{2}$ relative to the moving walkway. (a) How long does ir take the woman to travel the distance $L_{2}^{2}$ (b) How long does it take the man to travel this distance?

Ryan Williams

Numerade Educator

Problem 45

How long does it take an automobile traveling in the left lane of a highway at $60.0 \mathrm{~km} / \mathrm{h}$ to overtake (become even with) another car that is traveling in the right lane at $40.0 \mathrm{~km} / \mathrm{h}$ when the cars' front bumpers are initially $100 \mathrm{~m}$ apart?

Averell Hause

Carnegie Mellon University

Problem 46

You can use any coordinate system you like to solve a projectile motion problem. To demonstrate the truth of this statement, consider a ball thrown off the top of a building with a velocity $\overrightarrow{\mathrm{v}}$ at an angle $\theta$ with respect to the horizontal. Let the building be $50.0 \mathrm{~m}$ tall, the initial horizontal velocity be $9.00 \mathrm{~m} / \mathrm{s}$, and the initial vertical velocity be $12.0 \mathrm{~m} / \mathrm{s}$. Choose your coordinates such Lhat the positive $y$ -axis is upward, the $x$ -axis is to the right, and the origin is at the point where the ball is released.
(a) With these choices, find the ball's maximum height above the ground and the time it takes to reach the maximum height. (b) Repeat your calculations choosing the origin at the base of the building.

Ryan Williams

Numerade Educator

Problem 47

A Nordic jumper goes off a ski jump at an angle of $10.0^{\circ}$ below the horizontal, traveling $108 \mathrm{~m}$ horizontally and $55.0 \mathrm{~m}$ vertically before landing. (a) Ignoring friction and aerodynamic effects, calculate the speed needed by the skier on leaving the ramp. (b) Olympic Nordic jumpers can make such jumps with a jump speed of $23.0 \mathrm{~m} / \mathrm{s}$, which is considerably less than the answer found in part (a). Explain how that is possible.

Supratim Pal

Numerade Educator

Problem 48

In a local diner, a customer slides an empty coffee cup down the counter for a refill. The cup slides off the counter and strikes the floor at distance $d$ from the base of the counter. If the height of the counter is $h .$ (a) find an expression for the time $t$ it takes the cup to fall to the floor in terms of the variables $h$ and $g$. (b) With what speed does the mug leave the counter? Answer in terms of the variables $d, g$, and $h .(c)$ In the same terms, what is the speed of the cup immediately before it hits the floor?
(d) In terms of $h$ and $d$, what is the direction of the cup's velocity immediately before it hits the floor?

Averell Hause

Carnegie Mellon University

Problem 49

Towns $A$ and $B$ in Figure $P 3.49$ are $80.0 \mathrm{~km}$ apart. $A$ couple arranges to drive from town $A$ and meet a couple driving from town $\mathrm{B}$ at the lake, $\mathrm{L}$. The two couples leave simultaneously and drive for $2.50 \mathrm{~h}$ in the directions shown. Car 1 has a speed of $90.0 \mathrm{~km} / \mathrm{h}$. If the cars arrive simultaneously at the lake, what is the speed of car 2 ?

Averell Hause

Carnegie Mellon University

Problem 50

A chinook salmon has a maximum underwater speed of $3.58 \mathrm{~m} / \mathrm{s}$, but it can jump out of water with a speed of $6.26 \mathrm{~m} / \mathrm{s}$. To move upstream past a waterfall. the salmon does not need to jump to the top of the fall, but only to a point in the fall where the water speed is less than $3.58 \mathrm{~m} / \mathrm{s}$; it can then swim up the fall for the remaining distance. Because the salmon must make forward progress in the water, let's assume it can swim to the top if the water speed is $3.00 \mathrm{~m} / \mathrm{s}$. If water has a speed of $1.50 \mathrm{~m} / \mathrm{s}$ as it passes over a ledge, how far below the ledge will the water be moving with a speed of $3.00 \mathrm{~m} / \mathrm{s}$ ? (Note that water undergoes projectile motion once it leaves the ledge.) If the salmon is able to jump vertically upward from the base of the fall, what is the maximum height of waterfall that the salmon can clear?

Donald Albin

Numerade Educator

Problem 51

A rocket is launched at an angle of $53.0^{\circ}$ above the horizontal with an initial speed of $100 \mathrm{~m} / \mathrm{s}$. The rocket moves for $3.00 \mathrm{~s}$ along its initial line of motion with an acceleration of $30.0 \mathrm{~m} / \mathrm{s}^{2}$. At this time, its engines fail and the rocket proceeds to mose as a projectile. Find (a) the maximum altitude reached by the rocket, (b) its toral time of flight, and (c) its horizontal range.

Mark J

Numerade Educator

Problem 52

Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relaLive to the water, One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction. an observer on shore determines the velocities of the two canoes to be $-1.2 \mathrm{~m} / \mathrm{s}$ and $+2.9 \mathrm{~m} / \mathrm{s}$, respectively.
(a) What is the speed of the water relative to the shore?
(b) What is the speed of each canoe relative to the water?

Ryan Williams

Numerade Educator

Problem 53

If a person can jump a maximum horizontal distance (by using a 45 projection angle) of $3.0 \mathrm{~m}$ on Earth, what would be his maximum range on the Moon, where the free-fall acceleration is $g / 6$ and $g=9.80 \mathrm{~m} / \mathrm{s}^{2} ?$ Repeat for Mars, where the acceleration due to gravity is $0.38 g$.

Ryan Williams

Numerade Educator

Problem 54

$A$ daredevil decides to jump a canyon. Its walls are equally high and $10 \mathrm{~m}$ apart. He takes off by driving a motorcycle up a short ramp sloped at an angle of $15^{\circ} .$ What minimum speed must he have in order to clear the canyon?

Ryan Williams

Numerade Educator

Problem 55

A home run is hit in such a way that the baseball just clears a wall $21 \mathrm{~m}$ high, loeated $130 \mathrm{~m}$ from home plate. The ball is hit at an angle of $35^{\circ}$ to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of $1.0 \mathrm{~m}$ above the ground.)

Donald Albin

Numerade Educator

Problem 56

A ball is thrown straight upward and returns to the thrower's hand after $3.00 \mathrm{~s}$ in the air. A second ball is thrown at an angle of $30.0^{\circ}$ with the horizontal. At what speed must the second ball be thrown so that it reaches the same height as the one thrown vertically?

Narayan Hari

Numerade Educator

Problem 57

$A$ quarterback throws a football toward a receiver with an initial speed of $20 \mathrm{~m} / \mathrm{s}$ at an angle of $30^{\circ}$ above the horizontal. At that instant the receiver is $20 \mathrm{~m}$ from the quarterback. In what direction and with what constant speed should the receiver run in order to catch the football at the level at which it was thrown?

Matthew Baker

Numerade Educator

Problem 58

A $2.00-\mathrm{m}$ -tall basketball player is standing on the floor $10.0 \mathrm{~m}$ from the basket, as in Figure $\mathrm{P} 3.58$. If he shoots the ball at a $40.0^{\circ}$ angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is $3.05 \mathrm{~m}$

Averell Hause

Carnegie Mellon University

Problem 59

ecp In a very popular lecture demonstration, a projectile is fired at a falling target as in Figure $\mathrm{P} 3.59 .$ The projectile leaves the gun at the same instant the target is dropped from rest. Assuming the gun is initially aimed at the target, show that the projectile will hit the target. (One restriction of this experiment is that the projectile must reach the target before the target strikes the floor.)

Donald Albin

Numerade Educator

Problem 60

Figure P3.60 illustrates the difference in proportions between the male (m) and female (f) anatomies. The displacements $\overrightarrow{\mathbf{d}}_{1 \mathrm{~m}}$ and $\overrightarrow{\mathbf{d}}_{1 r}$ from the bottom of the feet to the navel have magnitudes of $104 \mathrm{~cm}$ and $84.0 \mathrm{~cm}$, respectively. The displacements $\overrightarrow{\mathrm{d}}_{2 \mathrm{ng}}$ and $\overrightarrow{\mathrm{d}}_{2 \text { f have magni- }}$ tudes of $50.0 \mathrm{~cm}$ and $43.0 \mathrm{~cm}$, respectively, (a) Find the vector sum of the displacements $\overrightarrow{\mathrm{d}}_{\mathrm{d} 1}$ and $\overrightarrow{\mathrm{d}}_{\mathrm{t} 2}$ in each case.
(b) The male figure is $180 \mathrm{~cm}$ tall, the female $168 \mathrm{~cm}$. Normalize the displacements of each figure to a common height of $200 \mathrm{~cm}$ and re-form the vector sums as in part (a). Then find the vector difference between the two
sums.

Donald Albin

Numerade Educator

Problem 61

By throwing a ball at an angle of $45^{\circ}$, a girl can throw the ball a maximum horizontal distance $R$ on a level field. How far can she throw the same ball vertically upward? Assume her muscles give the ball the same speed in each case. (Is this assumption valid?)

Donald Albin

Numerade Educator

Problem 62

The equation of a parabola is $y=a x^{2}+b x+c$, where $a, b$, and $c$ are constants. The $x$ - and $y$ -coordinates of a projectile launched from the origin as a function of time are given by $x=v_{0 x} l$ and $y=v_{10} l-\frac{1}{2} g t^{2}$, where $v_{0 x}$ and $v_{3}$ are the components of the initial velocity. (a) Eliminate 4 from these two equations and show that the path of a projectile is a parabola and has the form $y=a x+b x^{2}$.
(b) What are the values of $a, b$, and $c$ for the projectile?

Donald Albin

Numerade Educator

Problem 63

A hunter wishes to cross a river that is $1.5 \mathrm{~km}$ wide and flows with a speed of $5.0 \mathrm{~km} / \mathrm{h}$ parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of $12 \mathrm{~km} / \mathrm{h}$ with respect to the water. What is the minimum time necessary for crossing?

Averell Hause

Carnegie Mellon University

Problem 64

lecp When baseball outfielders throw the ball, they usually allow it to take one bounce, on the theory that the ball arrives at its target sooner that way. Suppose that. after the bounce, the ball rebounds at the same angle $\theta$ that it had when it was released (as in Fig. P3.64), but loses half its speed. (a) Assuming that the ball is always thrown with the same initial speed, at what angle $\theta$ should the ball be thrown in order to go the same distance $D$ with one bounce as a ball thrown upward at $45.0^{\circ}$ with no bounce?

Donald Albin

Numerade Educator

Problem 65

A daredevil is shot out of a cannon at $45.0^{\circ}$ to the horizontal with an initial speed of $25.0 \mathrm{~m} / \mathrm{s}$. A net is positioned a horizontal distance of $50.0 \mathrm{~m}$ from the cannon. At what height above the cannon should the net be placed in order to catch the daredevil?

Averell Hause

Carnegie Mellon University

Problem 66

Chinook salmon are able to move upstream faster by jumping out of the water periodically; this behavior is called porpoising. Suppose a salmon swimming in still water jumps out of the water with a speed of $6.26 \mathrm{~m} / \mathrm{s}$ at an angle of $45^{\circ}$, sails through the air a distance $L$ before returning to the water, and then swims a distance $L$. underwater at a speed of $3.58 \mathrm{~m} / \mathrm{s}$ before beginning another porpoising maneuver. Determine the average speed of the fish.

Averell Hause

Carnegie Mellon University

Problem 67

A student decides to measure the muzzle velocity of a pellet shot from his gun. He points the gun horizontally. He places a target on a vertical wall a distance $x$ away from the gun. The pellet hits the target a vertical distance y below the gun. (a) Show that the position of the pellet when traveling through the air is given by $y=A x^{2}$, where $A$ is a constant. (b) Express the constant $A$ in terms of the initial (muzzle) velocity and the free-fall acceleration. (c) If $x=3.00 \mathrm{~m}$ and $y=0.210 \mathrm{~m}$, what is the initial speed of the pellet?

Donald Albin

Numerade Educator

Problem 68

$A$ sailboat is heading directly north at a speed of $20 \mathrm{knots}(\mathrm{I} \mathrm{knot}=0.514 \mathrm{~m} / \mathrm{s})$. The wind is blowing
towards the east with a speed of 17 knots. Determine the magnitude and direction of the wind velocity as measured on the boat. What is the component of the wind velocity in the direction parallel to the motion of the boat? (See Problem $4.58$ for an explanation of how a sailboat can move "into the wind.")

Donald Albin

Numerade Educator

Problem 69

A golf ball with an initial speed of $50.0 \mathrm{~m} / \mathrm{s}$ lands exactly $240 \mathrm{~m}$ downrange on a level course. (a) Neglecting air friction, what tao projection angles would achieve this result? (b) What is the maximum height reached by the ball, using the two angles determined in part (a)?

Donald Albin

Numerade Educator

Problem 70

ecp $A$ landscape architect is planning an artificial waterfall in a city park. Water flowing at $0.750 \mathrm{~m} / \mathrm{s}$ leaves the end of a horizontal channel at the top of a vertical wall $9.35 \mathrm{~m}$ high and falls into a pool. (a) How far from the wall will the water land? Will the space behind the waterfall be wide enough for a pedestrian walkway? (b) To sell her plan to the city council, the architect wants to build a model to standard scale, one-twelfth actual size. How fast should the water flow in the channel in the model?

Donald Albin

Numerade Educator

Problem 71

One strategy in a snowball fight is to throw a snowball at a high angle over level ground. Then, while your opponent is watching that snowball, you throw a second one at a low angle timed to arrive before or at the same time as the first one. Assume both snowballs are thrown with a speed of $25.0 \mathrm{~m} / \mathrm{s}$. The first is thrown at an angle of $70.0^{\circ}$ with respect to the horizontal. (a) At what angle should the second snowball be thrown to arrive at the same point as the first? (b) How many seconds later should the second snowball be thrown after the first in order for both to
arrive at the same time?

Manish Jain

Numerade Educator

Problem 72

A dart gun is fired while being held horizontally at a height of $1.00 \mathrm{~m}$ above ground level and while it is at rest relative to the ground. The dart from the gun travels a horizontal distance of $5.00 \mathrm{~m}$. A college student holds the same gun in a horizontal position while sliding down a $45.0^{\circ}$ incline at a constant speed of $2.00 \mathrm{~m} / \mathrm{s}$. How far will the dart travel if the student fires the gun when it is $1.00 \mathrm{~m}$ above the ground?

Manish Jain

Numerade Educator

Problem 73

The determined Wile $\mathrm{E}$. Coyote is out once more to try to capture the elusive roadrunner. The coyote wears a new pair of Acme power roller skates, which provide a constant horizontal acceleration of $15 \mathrm{~m} / \mathrm{s}^{2}$, as shown in Figure P3.73. The coyote starts off at rest $70 \mathrm{~m}$ from the edge of a cliff at the instant the roadrunner zips by in the direction of the cliff. (a) If the roadrunner moves with constant speed, find the minimum speed the roadrunner must have to reach the cliff before the coyote. (b) If the cliff is $100 \mathrm{~m}$ above the base of a canyon, find where the coyote lands in the canyon. (Assume his skates are still in operation when he is in "flight" and that his horizontal component of acceleration remains constant at $15 \mathrm{~m} / \mathrm{s}^{2} .$.

Manish Jain

Numerade Educator